Behavior Of Bright-Dark Solitons Under Trapping Potentials Shreya Dilipkumar Shah In partial fulfilment of the requirements for the Degree of Master of Science Department of Physics, National University of Singapore 2010 Acknowledgement I am very much thankful to my supervisor, Prof. Li Baowen, Department of Physics, National University of Singapore, for giving me such a nice opportunity to work with him. His ideas, suggestions and encouragement have helped me a lot for doing research work and writing and preparing this project work. I am also very much thankful to Prof. Gong Jiangbin and Dr. Xiong Bo, Department of Physics, National University of Singapore as they have given good guidance for my project work. I am thankful to my friends Juzar and Bijay for helping me regarding computational work. I am also thankful to all my lab mates and dear friends for giving moral support during my stay at Singapore. I am very much thankful to my parents and loving brother Archit for supporting me in my Master Project. I am also very much thankful to my grandfather for believing in me. And at last how can I forget the Department of Physics, National University of Singapore for giving me the opportunity to do Master course. My thanks are also to all the faculty members of the department. Abstract We investigate the vector-soliton solutions of two-species Bose-Einstein condensates with arbitrary scattering lengths. The results show that, for quasi-one-dimension homogeneous systems, exact solutions exist even in the regime where the nonlinear system is non-integrable. In addition, these vector solitons can be dynamically stable in the presence of soft trapping potentials. If trapping potentials are strong, then these vector solitons can be dynamically unstable. In this case, Bright and Dark solitons will be destroyed. Index Chapter 1 Introduction ~1~ 3 Chapter 2 2.1 What is Soliton? 2.2 Solutions for Bright and Dark Solitons 2.3 6 8 Numerical Method to study Dynamics and Dynamical Stability for Bright and Dark Chapter 3 3.1 Solitons 12 Results and Discussion 14 Density Profiles for Different Trapping Potentials 16 3.2 Density Profiles for Special cases 54 3.3 Density Profiles for Different b12 values 61 Conclusion 74 Scope for the future work 75 Chapter 4 ~2~ Appendix A Gross-Pitaevskii Equation 75 Appendix B Thomas-Fermi Approximation 76 References 78 Chapter-1 ~3~ Introduction: A Bose-Einstein condensate (BEC) is a state of matter of a dilute gas of weakly interacting bosons confined in an external potential and cooled to temperatures very near to absolute zero (0 K or -273.15º C or -459.67 º F) [1]. Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the external potential, at which point quantum effects become apparent on a macroscopic scale. This state of matter was first predicted by Satyendra Nath Bose and Albert Einstein in 1925. Seventy years later, the first gaseous condensate was produced by Eric Cornell and Carl Wieman in 1995 at the University of Colorado at Boulder NIST-JILA lab, using a gas of Rubidium atoms cooled to 170 nanokelvin (nK). For their achievements Cornell, Wieman and Wolfgang Ketterle at MIT received the 2001 Noble Prize in Physics. Bose-Einstein Condensates in dilute atomic gases, which were first realized experimentally in 1995 for Rubidium [2], Sodium [3], and Lithium [4], provide unique opportunities for exploring quantum phenomena on a macroscopic scale. The particle density at the centre of a Bose-Einstein condensed atomic cloud is typically 1013-1015 cm-3. By contrast, the density of molecules in air at room temperature and atmospheric pressure is about 1019 cm-3. In liquids and solids the density is of order 1022 cm-3, while the density of nucleons in atomic nuclei is about 1038 cm-3. To observe quantum phenomena in such low-density systems, the temperature must be of order 10-5 K or less. This may be contrasted with the temperatures at which quantum phenomena occur in solids and liquids. In solids, quantum effects become strong for electrons in metals below the Fermi temperature, which is typically 104-105 K, and for phonons below the Debye temperature, which is typically of order 102 K. For the helium liquids, the temperatures required for observing quantum phenomena are of order 1 K. Due to the much higher particle density in atomic nuclei, the corresponding degeneracy temperature is about 1011 K. The path that led in 1995 to the first realization of Bose-Einstein condensation in dilute gases exploited the powerful methods developed over the past quarter of a century for cooling alkali metal atoms by using lasers. Since laser cooling alone cannot produce sufficiently high densities and low temperature for condensation, it is followed by an evaporative cooling stage, in which more energetic atoms are removed from the trap, thereby cooling the remaining atoms. Cold gas clouds have many advantages ~4~ for investigations of quantum phenomena. A major one is that in the Bose-Einstein condensate, essentially all atoms occupy the same quantum state, and the condensate may be described very well in terms of a mean-field theory. Although the gases are dilute, interactions play an important role because temperatures are so low, and they give rise to collective phenomena related to those observed in solids, quantum liquids and nuclei. Experimentally, the systems are attractive ones to work with, since they may be manipulated by the use of lasers and magnetic fields. In addition, interactions between atoms may be varied either by using different atomic species, or, for species that have a Feshbach resonance, by changing the strength of an applied magnetic or electric field. A further advantage is that, because of low density, ‘microscopic’ length scales are so large that the structure of the condensate wave function may be investigated directly by optical means. Finally, real collision processes play little role, and therefore these systems are ideal for studies of interference phenomena and atom optics. Since the first realization of BEC, a tremendous amount of research has been taken place in this interdisciplinary field [2, 3, 4, 5]. One specific topic of wide interest is multi-component BECs, which possess complicated quantum phases [6] and might provide a candidate model for quantum simulations. Theoretical [7-9] and experimental [10-12] studies have shown that inter-species interaction plays crucial roles in these systems. Recently two-species BECs with tunable interspecies interaction were successfully produced [13, 14]. This important progress motivates further explorations to understand and utilize the peculiar of two-species BECs. One important aspect of BEC dynamics is matter wave solitons described by nonlinear Schrodinger equations [5, 15-17]. Solitons represent a highly nonlinear wave phenomenon with unique propagation features and have attracted great interests from many different fields, e.g., nonlinear optics [18]. Because the equations of motion for matter-wave solitons are similar to those for solitons in other contexts, well-established understanding of solitons can be applied to BEC systems to achieve better control of matter waves. In return, BEC systems make it possible to study the formation of solitons under some circumstances that cannot be realized in other field. In my project work, interesting behaviour of vector-soliton in two-species condensates with arbitrary scattering lengths has been studied. It is found that exact solutions exist even in the non-integrable regime and maintain their dynamic stability in the presence of soft trapping potentials. Density profiles for Bright and Dark solitons have been obtained. My project work contains four chapters. In first chapter, general introduction for BECs has been given. Numerical method to study density profile for Bright and Dark solitons has been given in second ~5~ chapter. Third chapter contains results for various cases having different trapping potentials. And fourth chapter will show conclusion, limitations and future scope of this work. Chapter-2 ~6~ 2.1 What is Soliton? A soliton is a self-reinforcing solitary wave (a wave packet or pulse) that maintains its shape while it travels at constant speed [19]. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. The term “dispersive effects” refers to a property of certain systems where the speed of the waves varies according to frequency. Solitons arise as the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems. The soliton phenomenon was first described by John Scott Russell who observed a solitary wave in the Union Canal in Scotland. He produced the phenomenon in a wave tank and named it the “Wave of Translation”. Thus, 1) Solitons are of permanent form; 2) Solitons are localised within a region; 3) Solitons can interact with other solitons, and emerge from the collision unchanged, except for a phase shift. ~7~ Bright and Dark are two types of solitons. Bright solitons are associated with self-focusing or selfattractive waves while Dark solitons are associated with self-defocusing or self-repulsive waves. In the case of self-focusing, the fundamental bright soliton has the following general form: 𝑢𝑢(𝑧𝑧, 𝑥𝑥) = 𝑎𝑎 sech⁡ [(𝑎𝑎(𝑥𝑥 − 𝑣𝑣𝑣𝑣)]exp⁡ [𝑖𝑖𝑖𝑖𝑖𝑖 + 𝑖𝑖(𝑎𝑎 2 − 𝑣𝑣 2 )𝑧𝑧 2 ] , --- (2.1) where, a is the soliton amplitude and ν is the velocity. For spatial solitons, ν represents the transverse velocity of solitons propagating at an angle to the z axis. In the case of self-defocusing, the fundamental dark soliton has the following general form: 𝑢𝑢(𝑧𝑧, 𝑥𝑥) = 𝑢𝑢0 {B tanh[𝑢𝑢0 B(x − A𝑢𝑢0 z)] + 𝑖𝑖𝑖𝑖} exp⁡ [−𝑖𝑖𝑢𝑢02 𝑧𝑧] , --- (2.2) where, the parameters A and B are connected by the relation A2 + B2 = 1. An important difference between the bright and dark solitons is that the speed of a dark soliton depends on its amplitude. In contrast with bright solitons, which have a constant phase, the phase of a dark soliton changes across its width. Thus, by analogy, the word dark is used to describe solitons that correspond to density depressions. This category of solitons is further divided into black ones, for which the minimum density is zero, and grey ones, for which it is greater than zero. Solitons with a density maximum are referred to as bright. Bright Soliton and Dark Soliton ~8~ 2.2 Solutions for Bright and Dark Solitons: Our main aim is to show the interesting behaviour of vector-soliton in two-species condensates with arbitrary scattering lengths [20]. It is found that exact solutions exist even in the non-integrable regime and maintain their dynamic stability in the presence of soft trapping potentials. Different types of vectorsolitons can be realized and transformed to each other when the atom-atom interaction strength is tuned via Feshbach resonance. This opens up new possibilities in the quantum control of multicomponent BECs and other related complex systems. The mean-field dynamics of a two-species BEC is governed by the following equations [reference]: iℏ iℏ �1 ∂Ψ ∂t �2 ∂Ψ ∂t = �− = �− −ℏ2 ∇2 2 m1 −ℏ2 ∇2 2 m2 + V1 + U11 |Ψ1 |2 + U12 |Ψ2 |2 � Ψ1 , + V2 + U21 |Ψ1 |2 + U22 |Ψ2 |2 � Ψ2 , --- (2.3) --- (2.4) where, the condensate wave functions are normalized by particle numbers Ni = ∫|Ψi |2 d3 r, and intra- and inter- species interaction strengths are given as, Uii = 4 π ℏ2 aii� mi , U12 = U21 = 2 π ℏ2 a12� m respectively, with aij being the corresponding scattering lengths and m being the reduced mass. The trapping potentials are assumed to be Vi = mi [ω2ix x 2 + ω2i𝖳𝖳 (y 2 + z 2 )]/2 , Further assuming ωiT >> ωix such that transverse motion of the condensates are frozen to the ground state of the transverse harmonic trapping potential, the system becomes quasi-one-dimensional. Integrating out the transverse coordinates, the resulting equations for the axial wave functions �1,2 (x) in dimensionless form can be written as, Ψ i i �1 ∂Ψ ∂t �2 ∂Ψ ∂t = �− = �− 1 ∂2 λ1 2 x 2 κ ∂2 λ2 2 x 2 2 ∂x 2 ∂x + 2 + 2 2 2κ �1 �2 + b12 �Ψ � 2 �2 � Ψ �1 , + b11 �Ψ �1 �2 + b22 �Ψ � 2 �2 � Ψ �2 , + b21 �Ψ Here, we have chosen �ℏ�(m ω ) and 1 1T 2 π� ω1T ~9~ --- (2.5) --- (2.6) to be the units for length and time, 2 � � dx = 1 and �1,2 is normalized such that ∫�Ψ respectively; and Ψ 1 2 � � dx = ∫�Ψ 2 N2� , N1 Other parameters in equations (2.5) and (2.6) are defined as following: b11 = 2 a11 N1 , N 1 b12 = b21 = 2 m1 a12 [(1+γ)m] , b22 = 2 m1 a22 γ = ω2T /ω1T , N1γ m2 , λ1 = ω1x /ω1T , λ2 = ω2x /ω1T , κ = m1 /m2 , When the longitudinal trapping potential is neglected (λi = 0), Equations (2.3) and (2.4) are perfectly integrable under the condition κ = 1 (i.e. m1 = m2) and b11 = b12 = b21 = b22 [21], allowing for a general procedure to construct two-component vector-soliton solutions in the form of “dark-dark” [22], “brightdark” [23] and “bright-bright” [24] solitons. When integrability is destroyed, different studies have shown that distorted versions of soliton solutions exist, but closed form can only be given for special cases. Remarkably, we show here that even when the above-mentioned integrability condition is violated, there still exists, in general a specific class of exact vector-soliton solutions for arbitrary interaction strengths, so long as b212 ≠b11b22. ~ 10 ~ The vector-soliton solutions can be found by inserting an appropriate ansatz into Equations (2.5) and (2.6). Defining the following two quantities: C1 ≡ (b22 – κ b12) / ( b212 - b11b22) , --- (2.7) C2 ≡ (b12 – κ b11) / ( b212 - b11b22) , --- (2.8) The conditions under which various vector-soliton solutions exist are found to be as following: C1 > 0, C2 < 0 : Bright – Bright (BB), C1 > 0, C2 > 0 : Bright – Dark (BD), C1 < 0, C2 > 0 : Dark – Dark (DD), C1 < 0, C2 < 0 : Dark – Bright (DB). Explicit expressions for these vector-solitons are then found from the ansatz we use. In particular, the BB solution (i.e., bright soliton for species 1 and bright soliton for species 2; analogous convention applies to all other solutions) is given by, �η2 − v2 �t �1B = η �C1 sech(ηx − ηvt) ei(vx+ Ψ vx 2 ) v2 , �κη2 − κ �t � 2B = η �−C2 sech(ηx − ηvt) ei( κ + Ψ 2 --- (2.9a) ) , --- (2.9b) The BD solution is given by, �1B = η �C1 sech(ηx − ηvt) ei(vx+f1t) , Ψ � 2D = [iv�C2 /κ + η �C2 tanh(ηx − ηvt)] eif2t , Ψ with f1 = η2 − v 2 2 v2 --- (2.10a) --- (2.10b) v2 − b12 C2 (η2 + κ2 ) AND f2 = − b22 C2 (η2 + κ2 ) , The DD solutions is found to be, �1D = [iv�−C1 + η �−C1 tanh(ηx − ηvt)] eif1t , Ψ � 2D = [iv�C2 /κ + η �C2 tanh(ηx − ηvt)] eif2t , Ψ --- (2.11a) --- (2.11b) with f1 = −η2 − v 2 (−b11 C1 + b 12 C 2 μ2 ) AND f2 = −μ η2 − v 2 (−b21 C1 + b 22 C 2 μ2 ), ~ 11 ~ Finally, the DB solution can be obtained from the BD solution (2.8) by exchanging indices 1 and 2, and let C1,2 → - C1,2. In all these cases, the parameter η determines the width of the soliton and can be found by the �1,2 . The parameter ν gives the velocity of the soliton. normalization condition for Ψ Here, we take 7Li as species 1 and 23Na as species 2, with fixed and realistic intra-species interaction strength b11 < 0, b22 > 0 and variable inter-species interaction strength b12. Figure [1] depicts the “phase diagram” for different regimes of vector solitons as b12 is varied. As is indicated in Figure [1], the system supports BB (b12 < κ b11), BD (κ b11 < b12 < b22/κ) and DD (b12 > b22/κ) vector soliton, while the condition for DB soliton can never be satisfied for this particular 7Li-23Na system. Here we are trying to study dynamical stability for Bright-Dark (BD) solitons. Dynamical stability of the vector-soliton solutions can be confirmed by direct numerical simulations of Equations (2.5) and (2.6). In our simulations, a harmonic trapping potential is added along the longitudinal direction to make the simulations more realistic. The trapping potential is weak such that its variation across the soliton scale is negligible. Such a soft trapping potential spatially confines the solitons without affecting their ~ 12 ~ essential properties. For initial condition, we use the exact solution of Equation (2.10a) for the bright component, while we multiply Equation (2.10b) by a Thomas-Fermi profile to simulate the dark component (Appendix B). 2.3 Numerical Method to study Dynamics and Dynamical Stability for Bright and Dark Solitons: We use an explicit, unconditionally stable and spectrally accurate numerical method to solve the GPE (2.5) and (2.6) for dynamics of BEC. Due to the external trapping potential V1 and V2, the solution Ψ(x,t) of (2.5) and (2.6) decays to zero exponentially fast when |x|→ condition to truncate this problem. ∞. So we have to choose boundary Here, we will use Time-Splitting Pseudospectral method to study behaviour of Bright and Dark solitons. First we use Time-Splitting method. We choose a time step Δt > 0. For n = 0, 1, 2, ... from time t = tn = nΔt to t = tn+1 = tn + Δt, the GPEs (2.5) and (2.6) are solved in two splitting steps as following:  First, consider following equations: i �1 ∂Ψ = �− ∂t i �2 ∂Ψ ∂t 1 ∂2 2 ∂x = �− + 2 λ1 2 x 2 2 λ2 2 x 2 κ ∂2 2 ∂x �1 �2 + b12 �Ψ � 2 �2 � Ψ �1 , + b11 �Ψ + 2 2κ 2 2 �1 � + b22 �Ψ �2� � Ψ �2 , + b21 �Ψ --- (2.5) --- (2.6)  To decouple linearity and nonlinearity, we consider first kinetic energy terms: i �1 ∂Ψ i ∂t �2 ∂Ψ ∂t = �− �1 1 ∂2Ψ 2 ∂x 2 = �− �, �2 κ ∂2Ψ 2 ∂x 2 �,  Next we take potential energy and nonlinear terms as following: --- (2.12) --- (2.13) i �1 ∂Ψ i ∂t �2 ∂Ψ ∂t λ 2x2 =�1 2 λ 2x2 =�2 2κ �1 �2 + b12 �Ψ � 2 �2 � Ψ �1 , + b11 �Ψ 2 2 �1 � + b22 �Ψ �2� � Ψ �2 , + b21 �Ψ ~ 13 ~ --- (2.14) --- (2.15) �1 (𝑥𝑥, 𝑡𝑡)| and |𝛹𝛹 �2 (𝑥𝑥, 𝑡𝑡)| invariant in  For time t ∈ [tn, tn+1], Equations (2.14) and (2.15) leave |𝛹𝛹 �1 (𝑥𝑥, 𝑡𝑡) and 𝛹𝛹 �2 (𝑥𝑥, 𝑡𝑡) as, time t, and thus it can be integrated exactly and we can get 𝛹𝛹 2 2 �1 �2 + b12 �Ψ � 2 �2 � (t − t n )] , --- (2.16) �1 (𝑥𝑥, 𝑡𝑡) = 𝛹𝛹 �1 (𝑥𝑥 − 𝑡𝑡𝑛𝑛 ) 𝐸𝐸𝐸𝐸𝐸𝐸[−𝑖𝑖 �λ1 x + b11 �Ψ 𝛹𝛹 2 2 2 𝐴𝐴𝐴𝐴𝐴𝐴 �1 �2 + b22 �Ψ � 2 �2 � (t − t n )] , --- (2.17) �2 (𝑥𝑥, 𝑡𝑡) = 𝛹𝛹 �2 (𝑥𝑥 − 𝑡𝑡𝑛𝑛 ) 𝐸𝐸𝐸𝐸𝐸𝐸[−𝑖𝑖 �λ2 x + b21 �Ψ 𝛹𝛹 2κ  These equations (2.16) and (2.17) will be used to solve equations (2.12) and (2.13) for generating wave functions to evaluate Density profiles [25, 26] for Bright and Dark solitons. ~ 14 ~ Chapter - 3 Results and Discussion: In this chapter dynamics of vector solitons inside a longitudinal trap have been discussed. Consider following equations, i i �1 ∂Ψ ∂t �2 ∂Ψ ∂t = �− = �− 1 ∂2 λ1 2 x 2 κ ∂2 λ1 2 x 2 2 ∂x 2 ∂x + 2 + 2 2 2κ �1 �2 + b12 �Ψ � 2 �2 � Ψ �1 , + b11 �Ψ �1 �2 + b22 �Ψ � 2 �2 � Ψ �2 , + b21 �Ψ --- (2.3) --- (2.4) for Bright and Dark solitons respectively. As shown in chapter 2, the Bright-Dark [BD] solution is given as, �1B = η �C1 sech(ηx − ηvt) ei(vx+f1t) , Ψ � 2D = [iv�C2 /κ + η �C2 tanh(ηx − ηvt)] eif2t , Ψ with f1 = η2 − v 2 2 v2 --- (2.8a) --- (2.8b) v2 − b12 C2 (η2 + κ2 ) AND f2 = − b22 C2 �η2 + κ2 � , To study the stability analysis for Bright-Dark solitons, we will change trapping potential λ1 and λ2 for Bright and Dark solitons respectively. Trapping potential for both solitons are not equal in all cases. While keeping λ2 constant for Dark soliton, λ1 will change having values 0, 0.2, 0.4, 0.6, 0.8 and 1.0. For example, if λ2 is zero and λ1 will go as 0, 0.2, 0.4, 0.6, 0.8 and 1.0 respectively. λ2 will be changed as 0, 0.2, 0.4, 0.6, 0.8 and 1.0. ~ 15 ~ 3.1: Density Profile for Different Trapping Potentials Here plots are shown for following six cases: 3.1.1: λ2 = 0 for λ1 = 0, 0.2, 0.4, 0.6, 0.8 and 1.0 3.1.2: λ2 = 0.2 for λ1 = 0, 0.2, 0.4, 0.6, 0.8 and 1.0 3.1.3: λ2 = 0.4 for λ1 = 0, 0.2, 0.4, 0.6, 0.8 and 1.0 3.1.4: λ2 = 0.6 for λ1 = 0, 0.2, 0.4, 0.6, 0.8 and 1.0 3.1.5: λ2 = 0.8 for λ1 = 0, 0.2, 0.4, 0.6, 0.8 and 1.0 3.1.6: λ2 = 1.0 for λ1 = 0, 0.2, 0.4, 0.6, 0.8 and 1.0 Here different density plots are shown for different values of λ1 and λ2. Also, some plots are shown for special cases for Bright and Dark solitons. As discussed in chapter 2, for Bright-Dark solitons, the condition is C1 > 0 and C2 > 0. To achieve this condition, values for different parameters are taken as b11 = -1.47, b22 = 5.89 and b12 = b21 = 2. These values are quite reasonable for getting Bright and Dark solitons as shown in the Phase Diagram in chapter 2. Velocity ν is taken as 0.1. In figures, upper figure shows density profile for Bright soliton while lower figure shows density profile for Dark soliton. ~ 16 ~ 3.1.1.1 Density Profile for λ1 = 0 and λ2 = 0 ~ 17 ~ 3.1.1.2 Density Profile for λ1 = 0.02 and λ2 = 0 ~ 18 ~ 3.1.1.3 Density Profile for λ1 = 0.04 and λ2 = 0 ~ 19 ~ 3.1.1.4 Density Profile for λ1 = 0.06 and λ2 = 0 ~ 20 ~ 3.1.1.5 Density Profile for λ1 = 0.08 and λ2 = 0 ~ 21 ~ 3.1.1.6 Density Profile for λ1 = 0.1 and λ2 = 0 ~ 22 ~ 3.1.2.1 Density Profile for λ1 = 0 and λ2 = 0.02 ~ 23 ~ 3.1.2.2 Density Profile for λ1 = 0.02 and λ2 = 0.02 ~ 24 ~ 3.1.2.3 Density Profile for λ1 = 0.04 and λ2 = 0.02 ~ 25 ~ 3.1.2.4 Density Profile for λ1 = 0.06 and λ2 = 0.02 ~ 26 ~ 3.1.2.5 Density Profile for λ1 = 0.08 and λ2 = 0.02 ~ 27 ~ 3.1.2.6 Density Profile for λ1 = 0.1 and λ2 = 0.02 ~ 28 ~ 3.1.3.1 Density Profile for λ1 = 0 and λ2 = 0.04 ~ 29 ~ 3.1.3.2 Density Profile for λ1 = 0.02 and λ2 = 0.04 ~ 30 ~ 3.1.3.3 Density Profile for λ1 = 0.04 and λ2 = 0.04 ~ 31 ~ 3.1.3.4 Density Profile for λ1 = 0.06 and λ2 = 0.04 ~ 32 ~ 3.1.3.5 Density Profile for λ1 = 0.08 and λ2 = 0.04 ~ 33 ~ 3.1.3.6 Density Profile for λ1 = 0.1 and λ2 = 0.04 ~ 34 ~ 3.1.4.1 Density Profile for λ1 = 0 and λ2 = 0.06 ~ 35 ~ 3.1.4.2 Density Profile for λ1 = 0.02 and λ2 = 0.06 ~ 36 ~ 3.1.4.3 Density Profile for λ1 = 0.04 and λ2 = 0.06 ~ 37 ~ 3.1.4.4 Density Profile for λ1 = 0.06 and λ2 = 0.06 ~ 38 ~ 3.1.4.5 Density Profile for λ1 = 0.08 and λ2 = 0.06 ~ 39 ~ 3.1.4.6 Density Profile for λ1 = 0 .1 and λ2 = 0.06 ~ 40 ~ 3.1.5.1 Density Profile for λ1 = 0 and λ2 = 0.08 ~ 41 ~ 3.1.5.2 Density Profile for λ1 = 0.02 and λ2 = 0.08 ~ 42 ~ 3.1.5.3 Density Profile for λ1 = 0.04 and λ2 = 0.08 ~ 43 ~ 3.1.5.4 Density Profile for λ1 = 0.06 and λ2 = 0.08 ~ 44 ~ 3.1.5.5 Density Profile for λ1 = 0.08 and λ2 = 0.08 ~ 45 ~ 3.1.5.6 Density Profile for λ1 = 0.1 and λ2 = 0.08 ~ 46 ~ 3.1.6.1 Density Profile for λ1 = 0 and λ2 = 0.1 ~ 47 ~ 3.1.6.2 Density Profile for λ1 = 0.02 and λ2 = 0.1 ~ 48 ~ 3.1.6.3 Density Profile for λ1 = 0.04 and λ2 = 0.1 ~ 49 ~ 3.1.6.4 Density Profile for λ1 = 0.06 and λ2 = 0.1 ~ 50 ~ 3.1.6.5 Density Profile for λ1 = 0.08 and λ2 = 0.1 ~ 51 ~ 3.1.6.6 Density Profile for λ1 = 0.1 and λ2 = 0.1 ~ 52 ~ Thus, as shown above, we can conclude for different density profiles as following:  When λ1 and λ2 are zero, i.e. there is no trapping potentials, there is a small oscillation or soliton is still moving as velocity is not zero. Soliton is moving slowly with constant velocity.  When λ2 is constant and λ1 is changing, λ2 follows λ1 such that as λ1 increases, we will get periodic oscillations as λ1 approaches form 0 to 1.0.  When λ2 > λ1 (slightly greater), periodic behaviour does not exist there. Part 3.2 : Density profile for Special Cases ~ 53 ~ Here we have shown some special cases for Bright and Dark solitons. For example, if velocity ν , trapping potentials λ1 = λ2 = 0 and inter-species Interaction strength b12 = 0 then what will be the behaviour of solitons. And we have got interesting results for such cases as following:  When λ1 = λ2 = 0, ν = 0 and b12 = 0, in this case, soliton is not moving as velocity is zero.  When λ1 = λ2 = 0.1, ν = 0 and b12 = 2, still soliton is stationary.  When λ1 = λ2 = 0.1, ν = 0.1 and b12 = 0, small oscillations will be there.  But when λ1 = λ2 = 1.0, ν = 0.1 and b12 = 0, [figure 3.7.5] Bright soltion is not moving as trapping potential is quite strong to trap the soliton. In this case, trapping potential is higher than kinetic energy.  In case of λ1 = λ2 = 0.1, ν = 0.5 and b12 = 2, i.e., velocity is increased, we will get large oscillations as shown in figure 3.7.6.  Plots are shown as following: ~ 54 ~ 3.2.1 Density profile for λ1 = λ2 = 0 with b12 = 2 and ν = 0 ~ 55 ~ 3.2.2 Density profile for λ1 = λ2 = 0.1 with b12 = 2 and ν = 0 ~ 56 ~ 3.2.3 Density profile for λ1 = λ2 = 0 with b12 = 0 and ν = 0 ~ 57 ~ 3.2.4 Density profile for λ1 = λ2 = 0.1 with b12= 0 and ν = 0.1 ~ 58 ~ 3.2.5 Density profile for λ1 = λ2 = 1 with b12 = 0 and ν = 0.1 ~ 59 ~ 3.2.6 Density profile for λ1 = λ2 =.0.1 with b12 = 2 and ν = 0.5 Part 3: Density Profile for Different b12 values ~ 60 ~ In this part, different values for b12 have been taken to evaluate density profiles. As shown in Phase Diagram in Chapter 2, Bright-Dark solitons exist in the range from -0.45 to 19.35. So to check this criterion, I have taken 10 values for b12 in the range from -0.45 to 19.35 and evaluated density profiles for Bright-Dark solitons as shown in the following figures: ~ 61 ~ 3.3.1 Density profile for λ1 = λ2 =.0.1 with b12 = -0.45 and ν = 0.1 ~ 62 ~ 3.3.2 Density profile for λ1 = λ2 =.0.1 with b12 = 1.53 and ν = 0.1 ~ 63 ~ 3.3.3 Density profile for λ1 = λ2 =.0.1 with b12 = 3.51 and ν = 0.1 ~ 64 ~ 3.3.4 Density profile for λ1 = λ2 =.0.1 with b12 = 5.49 and ν = 0.1 ~ 65 ~ 3.3.5 Density profile for λ1 = λ2 =.0.1 with b12 = 7.47 and ν = 0.1 ~ 66 ~ 3.3.6 Density profile for λ1 = λ2 =.0.1 with b12 = 9.45 and ν = 0.1 ~ 67 ~ 3.3.7 Density profile for λ1 = λ2 =.0.1 with b12 = 11.43 and ν = 0.1 ~ 68 ~ 3.3.8 Density profile for λ1 = λ2 =.0.1 with b12 = 13.41 and ν = 0.1 ~ 69 ~ 3.3.9 Density profile for λ1 = λ2 =.0.1 with b12 = 15.39 and ν = 0.1 ~ 70 ~ 3.3.10 Density profile for λ1 = λ2 =.0.1 with b12= 17.37 and ν = 0.1 ~ 71 ~ 3.3.11 Density profile for λ1 = λ2 =.0.1 with b12 = 19.35 and ν = 0.1 Thus, we can conclude that, ~ 72 ~  At b12 = -0.45, there is a Bright soliton with small oscillation but there is no stable Dark soliton at b12 = -0.45.  As b12 increases and going towards 19.35, there are Bright and Dark solitons but the thickness of oscillations is decreasing as shown in figures from 3.3.3 and 3.3.4.  But after that the oscillations start disappearing as b12 increases as shown in figures 3.3.5 and 3.3.6.  And finally for b12 = 19.35, there is no existence of Bright and Dark solitons.  This part gives the clear proof for b12 values responsible for Bright and Dark solitons. Chapter-4 ~ 73 ~ Conclusion:  The presence of the weak longitudinal trap causes the vector-soliton to oscillate while maintaining the overall shape as shown in figures for density profiles for Bright and Dark solitons.  The two coupled cubic nonlinear Schrodinger equations (2.5) and (2.6) in the absence of the trapping potential terms, support a special class of vector-soliton solutions even when the equations are not integrable.  Trapping potential plays an important role for the stability of Bright and Dark solitons.  Velocity is also key parameter for solitons to maintain their shapes while oscillating.  Inter-species interaction strength b12 is responsible for types of solitons. Types of solitons also change if b12 changes. For example, as shown in phase diagram, there are three regions related to b12 values so if it changes, we have Bright-Bright solitons, Bright-Dark solitons and Dark-Dark solitons respectively. Scope for the future work: ~ 74 ~  Here I have only studied dynamics for Bright-Dark solitons as time is very limited for me to finish my project work.  I would like to do my further study for stability analysis for Dark-Dark solitons and Bright-Bright solitons.  Soliton conversion via Feshbach resonance is also very interesting case so I would like to study in this area. ~ 75 ~ Appendix A: Gross-Pitaevskii Equation: The Gross-Pitaevskii Equation (GPE) describes the ground state of a quantum system of identical bosons using the Hartree-Fock approximation and the pseudopotential interaction model. In the Hartree-Fock approximation the total wave function Ψ of the system of N bosons is taken as a product of single-particle functions Ψ, Ψ (r1, r2..., rN) = ψ(r1) ψ(r2)...ψ(rN) , --- (A.1) where ri is the coordinate of the boson number i. The pseudopotential model Hamiltonian of the system is given as, ℏ2 H = ∑Ni=1(− 2m ∂2 ∂r 2i + V(ri )) + ∑i[...]... conditions under which various vector-soliton solutions exist are found to be as following: C1 > 0, C2 < 0 : Bright – Bright (BB), C1 > 0, C2 > 0 : Bright – Dark (BD), C1 < 0, C2 > 0 : Dark – Dark (DD), C1 < 0, C2 < 0 : Dark – Bright (DB) Explicit expressions for these vector -solitons are then found from the ansatz we use In particular, the BB solution (i.e., bright soliton for species 1 and bright soliton... , κ = m1 /m2 , When the longitudinal trapping potential is neglected (λi = 0), Equations (2.3) and (2.4) are perfectly integrable under the condition κ = 1 (i.e m1 = m2) and b11 = b12 = b21 = b22 [21], allowing for a general procedure to construct two-component vector-soliton solutions in the form of dark- dark” [22], “brightdark” [23] and bright- bright” [24] solitons When integrability is destroyed,...~8~ 2.2 Solutions for Bright and Dark Solitons: Our main aim is to show the interesting behaviour of vector-soliton in two-species condensates with arbitrary scattering lengths [20] It is found that exact solutions exist even in the non-integrable regime and maintain their dynamic stability in the presence of soft trapping potentials Different types of vectorsolitons can be realized and transformed... shown for different values of λ1 and λ2 Also, some plots are shown for special cases for Bright and Dark solitons As discussed in chapter 2, for Bright- Dark solitons, the condition is C1 > 0 and C2 > 0 To achieve this condition, values for different parameters are taken as b11 = -1.47, b22 = 5.89 and b12 = b21 = 2 These values are quite reasonable for getting Bright and Dark solitons as shown in the... density profile for Bright soliton while lower figure shows density profile for Dark soliton ~ 16 ~ 3.1.1.1 Density Profile for λ1 = 0 and λ2 = 0 ~ 17 ~ 3.1.1.2 Density Profile for λ1 = 0.02 and λ2 = 0 ~ 18 ~ 3.1.1.3 Density Profile for λ1 = 0.04 and λ2 = 0 ~ 19 ~ 3.1.1.4 Density Profile for λ1 = 0.06 and λ2 = 0 ~ 20 ~ 3.1.1.5 Density Profile for λ1 = 0.08 and λ2 = 0 ~ 21 ~ 3.1.1.6 Density Profile for... for Bright- Dark (BD) solitons Dynamical stability of the vector-soliton solutions can be confirmed by direct numerical simulations of Equations (2.5) and (2.6) In our simulations, a harmonic trapping potential is added along the longitudinal direction to make the simulations more realistic The trapping potential is weak such that its variation across the soliton scale is negligible Such a soft trapping. .. Bright and Dark solitons respectively As shown in chapter 2, the Bright- Dark [BD] solution is given as, �1B = η �C1 sech(ηx − ηvt) ei(vx+f1t) , Ψ � 2D = [iv�C2 /κ + η �C2 tanh(ηx − ηvt)] eif2t , Ψ with f1 = η2 − v 2 2 v2 - (2.8a) - (2.8b) v2 − b12 C2 (η2 + κ2 ) AND f2 = − b22 C2 �η2 + κ2 � , To study the stability analysis for Bright- Dark solitons, we will change trapping potential λ1 and λ2 for Bright. .. potential spatially confines the solitons without affecting their ~ 12 ~ essential properties For initial condition, we use the exact solution of Equation (2.10a) for the bright component, while we multiply Equation (2.10b) by a Thomas-Fermi profile to simulate the dark component (Appendix B) 2.3 Numerical Method to study Dynamics and Dynamical Stability for Bright and Dark Solitons: We use an explicit,... Bright and Dark solitons respectively Trapping potential for both solitons are not equal in all cases While keeping λ2 constant for Dark soliton, λ1 will change having values 0, 0.2, 0.4, 0.6, 0.8 and 1.0 For example, if λ2 is zero and λ1 will go as 0, 0.2, 0.4, 0.6, 0.8 and 1.0 respectively λ2 will be changed as 0, 0.2, 0.4, 0.6, 0.8 and 1.0 ~ 15 ~ 3.1: Density Profile for Different Trapping Potentials. .. method to solve the GPE (2.5) and (2.6) for dynamics of BEC Due to the external trapping potential V1 and V2, the solution Ψ(x,t) of (2.5) and (2.6) decays to zero exponentially fast when |x|→ condition to truncate this problem ∞ So we have to choose boundary Here, we will use Time-Splitting Pseudospectral method to study behaviour of Bright and Dark solitons First we use Time-Splitting method We choose ... vector solitons can be dynamically stable in the presence of soft trapping potentials If trapping potentials are strong, then these vector solitons can be dynamically unstable In this case, Bright. .. difference between the bright and dark solitons is that the speed of a dark soliton depends on its amplitude In contrast with bright solitons, which have a constant phase, the phase of a dark soliton changes... solutions in the form of dark- dark” [22], “brightdark” [23] and bright- bright” [24] solitons When integrability is destroyed, different studies have shown that distorted versions of soliton solutions